The problem of global exponential stabilization by boundary feedback for the Korteweg-de Vries-Burgers equation on the domain [0, 1] is considered. We derive a control law of the form u(0) = ux(1) = uxx(1) − k[u(1)3 + u(1)] = 0, where k is a sufficiently large positive constant, and prove that it guarantees L2-global exponential stability, H3-global asymptotic stability, and H3-semiglobal expon...

In [9] and [10], we have studied the initial value problem for the Korteweg-de Vries (KdV) equation (1.1) ut—6uu x +u xxx =0 by the inverse scattering method. In this paper we study the asymptotic behavior of the solutions as t— >zh°°-Consider the Schrodinger equation (1.2) over (— oo } oo) with the potential u(x) satisfying (1.3) (throughout the paper integration is taken over (— oo, oo) unles...

An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave U(x, t) under a stochastic time-dependent force is developed. The dynamics of the soliton wave U(x, t) is described by the Korteweg–de Vries (KdV) equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude . ...

We show that the quartic generalised KdV equation ut + uxxx + (u )x = 0 is globally wellposed for data in the critical (scale-invariant) space Ḣ −1/6 x (R) with small norm (and locally wellposed for large norm), improving a result of Gruenrock [8]. As an application we obtain scattering results in H x(R) ∩ Ḣ −1/6 x (R) for the radiation component of a perturbed soliton for this equation, improv...

In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ θ ≥ 1. Persistence property has been proved by approximation of the solutions and using a pri...

Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N -soliton solutions to our nonlocal equations correspond to self-adjoint reflectionless analytic difference operators with N bound states. A suitable scaling limit gives rise to the N -solito...

A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central force problem; inequivalent Lagrangians and Hamiltonians; constants of central force motion; a general discussion of higher-order Lagrangians and Hamiltonians with examples from Bohmian quantum mechanics, the Korteweg-de Vries equation and the ...

We prove special decay properties of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of pertu...

We find new analytic solitary-wave solutions, having a nonzero background at infinity, of the coupled Korteweg-De Vries equation, using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves can be stable or unstable, depending on the coefficients of the model. We study the interaction dynamics by usin...